Math 001 Homework 1

[pic 1]

MATH001 Calculus

Homework 1

HW1.Q1. [10 marks]. Solve the following equation, for 𝒙.  

𝟐𝒙 + 𝟗 = 𝟓(𝒙−𝟕)

SOLUTION:

2x + 9 = 5x – 35

9 + 35 = 5x – 2x

44 = 3x

x =  [pic 2]

HW1.Q2. [10 marks]. Solve the following equation, for 𝒙.  

𝟑𝒙𝟐 −𝒙 = 𝟐𝟐

SOLUTION:

0 = 3x2 – x – 22

Using this formula to solve quadratic equation [pic 3]

 [pic 4]

  [pic 5]

 [pic 6]

  [pic 7][pic 8]

HW1.Q3. [10 marks, Textbook, p. 101, P65, modified]. Solve the following equation, for 𝒕.  

𝟏. 𝟎𝟎𝟓𝟏𝟐𝒕 −𝟑 = 𝟎

SOLUTION:

1.00512t = 3

12t = log1.0053

t = [pic 9]

t =  [pic 10]

t = 18. 35594227 ≈ 18.36

HW1.Q4. [10 marks, Textbook, p. 100. P49]. Solve the following equation.  

𝐥og𝟏𝟎(𝒙−𝟏) −𝐥og𝟏𝟎(𝒙 + 𝟏) = 𝟏

SOLUTION:

Log10  = 1[pic 11]

  = 10[pic 12]

x -1 = 10(x+1)

x – 1 = 10x + 10

-1 -10 = 10x -x

-11 = 9x

x =  (no solution, log cannot be negative)[pic 13]

HW1.Q5. [ , Textbook, p. 65, P69]. At a price of $2.28 per bushel, the supply of barley is 7,500 million bushels and the demand is 7,900 million bushels. At a price of $2.37 per bushel, the supply is 7,900 million bushels and the demand is 7,800 million bushels. Let 𝒑 denote the price in dollars and let 𝒙 denote the number of bushels (in millions).

1. Find a price-supply equation of the form 𝒑 = 𝒎𝒙 + 𝒃.
2. Find a price-demand equation of the form 𝒑 = 𝒎𝒙 + 𝒃.
3. Find the equilibrium point.

SOLUTION:

a) p = mx + b

m =  =  [pic 14][pic 15]

m = 0.000225

if m = 0.000225, b = 2.37 – (0.000225 x 7900)

b = 0.5925

price-supply equation = p = 0.000225x + 0.5925

 b) p = mx + b

m =  =  [pic 16][pic 17]

m = -0.0009

if m = -0.0009, b = 2.37 – (-0.0009 x 7800)

b = 9.39

price-supply equation = p = -0.0009x + 9.39

c) 0.000225x + 0.5925 = -0.0009x + 9.39

0.0011225x = 8.7975

X = 7820

P = 0.000225(7820) + 0.5925

P = 2.352

Equilibrium point = (7820, 2.352)

HW1.Q6. [10 marks]. Use the following revenue function and cost function found by the marketing research department for a company that manufactures and sells memory chips for microcomputers,  

𝑹(𝒙) = 𝒙(𝟕𝟓−𝟑𝒙), 𝑪(𝒙)= 𝟏𝟐𝟓 + 𝟏6𝒙,

where 𝒙 is in millions of chips, and 𝑹(𝒙) and 𝑪(𝒙) are in millions of dollars. Both functions have domain 𝟏 ≤ 𝒙 ≤ 𝟐𝟎. Find the break-even point(s) to the nearest thousands chips.

SOLUTION:

Breakeven point = R(x) = C(x)

x(75 -3x) = 125 + 16x

75x – 3x2 = 125 + 16x

0 = 3x2 - 59x + 125 Using this formula to solve quadratic equation [pic 18]

 [pic 19]

  [pic 20]

x ≈ 17,251,000 or 2,415,000 (nearest thousand)

HW1.Q7. [10 marks, Textbook, p. 90, P61]. From the dawn of humanity to 1830, world population grew to one billion people. In 100 more years (by 1930) it grew to two billion, and 3 billion more were added in only 60 years (by 1990). In 2013, the estimated world population was 7.1 billion with a relative growth rate of 1.1%.